Multi-antenna techniques can significantly increase the data rates and reliability of a wireless communication system. The performance is in particular improved if both the transmitter and the receiver are equipped with multiple antennas, which results in a multiple-input multiple-output (MIMO) communication channel. Such systems and related techniques are commonly referred to as MIMO.
The 3GPP LTE standard is currently evolving with enhanced MIMO support. A core component in LTE is the support of MIMO antenna deployments and MIMO related techniques. A current working assumption in LTE-Advanced is the support of an 8-layer spatial multiplexing mode for 8 transmit (Tx) antennas, with the possibility of channel dependent precoding. The spatial multiplexing mode provides high data rates under favorable channel conditions.
With spatial multiplexing, an information carrying symbol vector s is multiplied by an NT×r precoder matrix WNT×r, which serves to distribute the transmit energy in a subspace of the NT (corresponding to NT antenna ports) dimensional vector space. The precoder matrix is typically selected from a codebook of possible precoder matrices, and typically indicated by means of a precoder matrix indicator (PMI). The PMI value specifies a unique precoder matrix in the codebook for a given number of symbol streams.
If the precoder matrix is confined to have orthonormal columns, then the design of the codebook of precoder matrices corresponds to a Grassmannian subspace packing problem. In any case, the r symbols in the symbol vector s each correspond to a layer and r is referred to as the transmission rank. In this way, spatial multiplexing is achieved because multiple symbols can be transmitted simultaneously over the same time/frequency resource element (TFRE). The number of symbols r is typically adapted to suit the current propagation channel properties.
LTE uses OFDM in the downlink (and DFT precoded OFDM in the uplink) and hence the received NR×1 vector yn for a certain TFRE on subcarrier n (or alternatively data TFRE number n) is thus modeled byyn=HnWNT×rsn+en  (1)where en is a noise/interference vector obtained as realizations of a random process. The precoder, WNT×r, can be a wideband precoder, which is constant over frequency, or frequency selective.
The precoder matrix is often chosen to match the characteristics of the NR×NT MIMO channel matrix H, resulting in so-called channel dependent precoding. This is also commonly referred to as closed-loop precoding and essentially tries to focus the transmit energy into a subspace which is strong in the sense of conveying much of the transmitted energy to the targeted receiver, e.g., a user equipment (UE). In addition, the precoder matrix also may be selected with the goal of orthogonalizing the channel, meaning that after proper linear equalization at the UE or other targeted receiver, the inter-layer interference is reduced.
In closed-loop precoding for the LTE downlink in particular, the UE transmits, based on channel measurements in the forward link (downlink), recommendations to the eNodeB of a suitable precoder to use. A single precoder that is supposed to cover a large bandwidth (wideband precoding) may be fed back. It also may be beneficial to match the frequency variations of the channel and instead feedback a frequency-selective precoding report, e.g. several precoders, one per frequency subband. This approach is an example of the more general case of channel state information (CSI) feedback, which also encompasses feeding back entities other than precoders, to assist the eNodeB in adapting subsequent transmissions to the UE. Such other information may include channel quality indicators (CQIs) as well as a transmission rank indicator (RI).
For the LTE uplink, the use of closed-loop precoding means that the eNodeB selects precoder(s) and the transmission rank. The eNodeB thereafter signals the selected precoder that the UE is supposed to use. The eNodeB also may use certain bitmap-based signaling to indicate the particular precoders within a codebook that the UE is restricted to use. See, e.g., Section 7.2 of the 3GPP Technical Specification, TS 36.213. One disadvantage of such signaling is the use of bitmaps to indicate allowed or disallowed precoders. Codebooks with large numbers of precoders require long bitmaps, and the signaling overhead associated with transmitting long bitmaps becomes prohibitive.
In any case, the transmission rank, and thus the number of spatially multiplexed layers, is reflected in the number of columns of the precoder. Efficiency and transmission performance are improved by selecting a transmission rank that matches the current channel properties. Often, the device selecting precoders is also responsible for selecting the transmission rank. One approach to transmission rank selection involves evaluating a performance metric for each possible rank and picking the rank that optimizes the performance metric. These kinds of calculations are often computationally burdensome and it is therefore an advantage if calculations can be re-used across different transmission ranks. Re-use of calculations is facilitated by designing the precoder codebook to fulfill the so-called rank nested property. This means that the codebook is such that there always exists a column subset of a higher rank precoder that is also a valid lower rank precoder.
The 4-Tx House Holder codebook for the LTE downlink is an example of a codebook that fulfills the rank nested property. The property is not only useful for reducing computational complexity, but is also important in simplifying overriding a rank selection at a device other than the one that has chosen the transmission rank. Consider for example the LTE downlink where the UE selects the precoder and rank, and conditioned on those choices, computes a CQI representing the quality of the effective channel formed by the selected precoder and the channel. Since the CQI thus reported by the UE is conditioned on a certain transmission rank, performing rank override at the eNodeB side makes it difficult to know how to adjust the reported CQI to take the new rank into account.
However, if the precoder codebook fulfills the rank nested property, overriding the rank to a lower rank precoder is possible by selecting a column subset of the original precoder. Since the new precoder is a column subset of the original precoder, the CQI tied to the original precoder gives a lower bound on the CQI if the new reduced rank precoder is used. Such bounds can be exploited for reducing the CQI errors associated with rank override, thereby improving the performance of the link adaptation.
Another issue to take into account when designing precoders is to ensure an efficient use of the transmitter's power amplifiers (PAs). Usually, power cannot be borrowed across antennas because, in general, there is a separate PA for each antenna. Hence, for maximum use of the PA resources, it is important that the same amount of power is transmitted from each antenna, i.e., a precoder matrix W should fulfill[WW*]mm=κ,∀m  (2)Thus, it is beneficial from a PA utilization point of view to enforce this constraint when designing precoder codebooks.
Full power utilization is also ensured by the so-called constant modulus property, which means that all scalar elements in a precoder have the same norm (modulus). It is easily verified that a constant modulus precoder also fulfills the full PA utilization constraint in (2) and hence the constant modulus property constitutes a sufficient but not necessary condition for full PA utilization.
As a further aspect of the LTE downlink and associated transmitter adaptation, the UE reports CQI and precoders to the eNodeB via a feedback channel. The feedback channel is either on the Physical Uplink Control Channel (PUCCH) or on the Physical Uplink Shared Channel (PUSCH). The former is a rather narrow bit pipe where CSI feedback is reported in a semi-statically configured and periodic fashion. On the other hand, reporting on PUSCH is dynamically triggered as part of the uplink grant. Thus, the eNodeB can schedule CSI transmissions in a dynamic fashion. Further, in contrast to CSI reporting on PUCCH, where the number of physical bits is currently limited to 20, CSI reports on PUSCH can be considerably larger. Such a division of resources makes sense from the perspective that semi-statically configured resources such as PUCCH cannot adapt to quickly changing traffic conditions, thus making it important to limit their overall resource consumption.
More generally, maintaining low signaling overhead remains an important design target in wireless systems. In this regard, precoder signaling can easily consume a large portion of the available resources unless the signaling protocol is carefully designed. The structure of possible precoders and the overall design of the precoder codebook plays an important role in keeping the signaling overhead low. A particularly promising precoder structure involves decomposing the precoder into two matrices, a so-called factorized precoder. The precoder can then be written as a product of two factorsWNT×r=WNT×k(c)Wk×r(t),  (3)where an NT×k conversion precoder WNT×k strives for capturing wideband/long-term properties of the channel such as correlation, while a k×r tuning precoder Wk×r(t) targets frequency-selective/short-term properties of the channel.
Together, the factorized conversion and tuning precoders represent the overall precoder WNT×r, which is induced by the signaled entities. The conversion precoder is typically, but not necessarily, reported with a coarser granularity in time and/or frequency than the tuning precoder to save overhead and/or complexity. The conversion precoder serves to exploit the correlation properties for focusing the tuning precoder in “directions” where the propagation channel on average is “strong.” Typically, this is accomplished by reducing the number of dimensions k covered by the tuning precoder. In other words, the conversion precoder WNT×k(c) becomes a tall matrix with a reduced number of columns. Consequently, the number of rows k of the tuning precoder Wk×r(t) is reduced as well. With such a reduced number of dimensions, the codebook for the tuning precoder, which easily consumes most of the signaling resources since it needs to be updated with fine granularity, can be made smaller while still maintaining good performance.
The conversion and the tuning precoders may each have a codebook of their own. The conversion precoder targets having high spatial resolution and thus a codebook with many elements, while the codebook the tuning precoder is selected from needs to be rather small in order to keep the signaling overhead at a reasonable level.
To see how correlation properties are exploited and dimension reduction achieved, consider the common case of an array with a total of NT elements arranged into NT/2 closely spaced cross-poles. Based on the polarization direction of the antennas, the antennas in the closely spaced cross-pole setup can be divided into two groups, where each group is a closely spaced co-polarized Uniform Linear Array (ULA) with NT/2 antennas. Closely spaced antennas often lead to high channel correlation and the correlation can in turn be exploited to maintain low signalling overhead. The channels corresponding to each such antenna group ULA are denoted H/ and H\, respectively. For convenience in notation, the following equations drop the subscripts indicating the dimensions of the matrices as well as the subscript n. Assuming now that the conversion precoder W(c) has a block diagonal structure,
                              W                      (            c            )                          =                              [                                                                                                      W                      ~                                                              (                      c                      )                                                                                        0                                                                              0                                                                                            W                      ~                                                              (                      c                      )                                                                                            ]                    .                                    (        4        )            The product of the MIMO channel and the overall precoder can then be written as
                                                        HW              =                                                [                                                                                                              H                          /                                                                                                                      H                          \                                                                                                      ]                                ⁢                                  W                                      (                    c                    )                                                  ⁢                                  W                                      (                    t                    )                                                                                                                          =                                                                    [                                                                                                                        H                            /                                                                                                                                H                            \                                                                                                                ]                                    ⁡                                      [                                                                                                                                                      W                              ~                                                                                      (                              c                              )                                                                                                                                0                                                                                                                      0                                                                                                                                    W                              ~                                                                                      (                              c                              )                                                                                                                                            ]                                                  ⁢                                  W                                      (                    t                    )                                                                                                                          =                                                [                                                                                                                                          H                            /                                                    ⁢                                                                                    W                              ~                                                                                      (                              c                              )                                                                                                                                                                                                        H                            \                                                    ⁢                                                                                    W                              ~                                                                                      (                              c                              )                                                                                                                                                            ]                                ⁢                                  W                                      (                    t                    )                                                                                                                          =                                                H                  eff                                ⁢                                                      W                                          (                      t                      )                                                        .                                                                                        (        5        )            As seen, the matrix {tilde over (W)}(c) separately precodes each antenna group ULA, thereby forming a smaller and improved effective channel Heff. As such, W(c) is sometimes referred to as an “antenna subgroup” precoder. If {tilde over (W)}(c) corresponds to a beamforming vector, the effective channel would reduce to having only two virtual antennas, which reduces the needed size of the codebook used for the second tuning precoder matrix W(t) when tracking the instantaneous channel properties. In this case, instantaneous channel properties are to a large extent dependent upon the relative phase relation between the two orthogonal polarizations.
It is also helpful for a fuller understanding of this disclosure to consider the theory regarding a “grid of beams,” along with Discrete Fourier Transform (DFT) based precoding. DFT based precoder vectors for NT transmit antennas can be written in the form
                                          w            n                          (                                                N                  T                                ,                Q                            )                                =                                    [                                                                                          w                                              1                        ,                        n                                                                    (                                                                              N                            T                                                    ,                          Q                                                )                                                                                                                        w                                              2                        ,                        n                                                                    (                                                                              N                            T                                                    ,                          Q                                                )                                                                                                  …                                                                              w                                                                        N                          T                                                ,                        n                                                                    (                                                                              N                            T                                                    ,                          Q                                                )                                                                                                        ]                        T                          ⁢                                  ⁢                                            w                              m                ,                n                                            (                                                      N                    T                                    ,                  Q                                )                                      =                          exp              ⁡                              (                                  j                  ⁢                                                            2                      ⁢                                                                                          ⁢                      π                                                                                      N                        T                                            ⁢                      Q                                                        ⁢                  mn                                )                                              ,                                          ⁢                      m            =            0                    ,          …          ⁢                                          ,                                    N              T                        -            1                    ,                      n            =            0                    ,          …          ⁢                                          ,                                    QN              T                        -            1                    ,                                    (        6        )            where wm,n(NT,Q) is the phase of the m:th antenna, n is the precoder vector index (i.e., which beam out of the QNT beams) and Q is the oversampling factor.
For good performance, it is important that the array gain function of two consecutive beams overlaps in the angular domain, so that the gain does not drop too much when going from one beam to another. Usually, this requires an oversampling factor of at least Q=2. Thus for NT antennas, at least 2NT beams are needed.
An alternative parameterization of the above DFT based precoder vectors is
                                          w                          l              ,              q                                      (                                                N                  T                                ,                Q                            )                                =                                    [                                                                                          w                                              1                        ,                                                  Ql                          +                          q                                                                                            (                                                                              N                            T                                                    ,                          Q                                                )                                                                                                                        w                                              2                        ,                                                  Ql                          +                          q                                                                                            (                                                                              N                            T                                                    ,                          Q                                                )                                                                                                  …                                                                              w                                                                        N                          T                                                ,                                                  Ql                          +                          q                                                                                            (                                                                              N                            T                                                    ,                          Q                                                )                                                                                                        ]                        T                          ⁢                                  ⁢                                            w                              m                ,                                  Ql                  +                  q                                                            (                                                      N                    T                                    ,                  Q                                )                                      =                          exp              ⁢                              (                                  j                  ⁢                                                            2                      ⁢                                                                                          ⁢                      π                                                              N                      T                                                        ⁢                                      m                    ⁡                                          (                                              l                        +                                                  q                          Q                                                                    )                                                                      )                                              ,                                    (        7        )            for m=0, . . . , NT−1, l=0, . . . , NT−1, q=0, 1, . . . , Q−1, and where l and q together determine the precoder vector index via the relation n=Ql+q. This parameterization also highlights that there are Q groups of beams, where the beams within each group are orthogonal to each other. The q:th group can be represented by the generator matrix
                              G          q                      (                          N              T                        )                          =                              [                                                                                w                                          0                      ,                      q                                                              (                                                                        N                          T                                                ,                        Q                                            )                                                                                                            w                                          1                      ,                      q                                                              (                                                                        N                          T                                                ,                        Q                                            )                                                                                        …                                                                      w                                                                                            N                          T                                                -                        1                                            ,                      q                                                              (                                                                        N                          T                                                ,                        Q                                            )                                                                                            ]                    .                                    (        8        )            By insuring that only precoder vectors from the same generator matrix are being used together as columns in the same precoder, it is straightforward to form sets of precoder vectors for use in so-called unitary precoding where the columns within a precoder matrix should form an orthonormal set.
Further, to maximize the performance of DFT based precoding, it is useful to center the grid of beams symmetrically around the broad size of the array. Such rotation of the beams can be done by multiplying from the left the above DFT vectors wn(NT,Q) with a diagonal matrix Wrot having elements
                                          [                          W              rot                        ]                    mm                =                              exp            ⁡                          (                              j                ⁢                                  π                                      QN                    T                                                  ⁢                m                            )                                .                                    (        9        )            The rotation can either be included in the precoder codebook or alternatively be carried out as a separate step where all signals are rotated in the same manner and the rotation can thus be absorbed into the channel from the perspective of the receiver (transparent to the receiver). Henceforth, in discussing DFT precoding herein, it is tacitly assumed that rotation may or may not have been carried out. That is, both alternatives are possible without explicitly having to mention it.
One aspect of the above-described factorized precoder structure relates to lowering the overhead associated with signaling the precoders, based on signaling the conversion and the tuning precoders W(c) and W(t) with different frequency and/or time granularity. The use of a block diagonal conversion precoder is specifically optimized for the case of a transmit antenna array consisting of closely spaced cross-poles, but other antenna arrangements exist as well. In particular, efficient performance with a ULA of closely spaced co-poles should also be achieved. However, the method for achieving efficient performance in this regard is not obvious, with respect to a block diagonal conversion precoder structure.
Another aspect to consider is that, in a general sense, the above-described factorized precoder feedback may prevent full PA utilization, and may violate the aforementioned rank nested property. These issues arise from the fact that the two factorized precoders—i.e., the conversion precoder and the tuning precoder—are multiplied together to form the overall precoder and thus the normal rules for ensuring full PA utilization and rank nested property by means of constant modulus and column subset precoders, respectively, do not apply.
Further precoding considerations, particularly in the context of the LTE downlink, include the fact that the PUCCH cannot bear as large a payload size as the PUSCH, for the previously described reasons. Thus, there is a risk of “coverage” problems when a UE reports CSI on the PUCCH. In this regard, it is useful to understand that current precoder designs commonly are optimized for transmissions to/from a single UE. In the MIMO context, this single-user context is referred to as a Single User MIMO or SU-MIMO. Conversely, co-scheduling multiple UEs on the same time/frequency resources is called Multi User MIMO or MU-MIMO. MU-MIMO is gaining increasing interest, but it imposes different requirements on precoder reporting and the underlying precoder structures.